Velocity

 

Since the relative velocities of objects must be considered in some of the examples that are considered, some general considerations about light, velocity, and time are discussed in this section.

 

Addition of velocities

 

The theory of the addition of velocities (TAV) can be derived from the Lorentz transformation. Consider two inertial reference frames, S and S’, with S’ moving with a velocity of v relative to S. Assume a point moving in S with a velocity of w relative to S’. The TAV is shown by Einstein[1] to be:

 

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W is the added value of the velocities. If the point is assumed to be a photon with a fixed velocity of c, then the formula shows that W = c. The only way that this can be true is if v = 0. To say that w can be equal to c, and in that case that W = c regardless of the value of v appears not to be correct under the assumed conditions. Light may be enigmatical, but it is not magical. Light can be expected to follow the ‘rules’ of the physical universe. The only logical interpretation of this is that if w = c then v must be equal to 0. Which is to say that if v > 0 then any point moving in S with a speed of w relative to S’ will not appear to go faster than w-v with respect to S’. So if w is the velocity of a light beam, then, setting w = c, its apparent speed relative to S’ must be c’ = c – v as seen from S [2]. The addition of velocities is considered under different conditions in a later section.

 

Velocity and Time

 

Special relativity considers that the clock rate in a reference frame changes as a function of the velocity of that frame. Some considerations that seem relevant to the issue will be discussed here.

 

Consider two systems, S and S’ that have some velocity v relative to each other. A device is in S’ that flashes periodically at some time interval t_d. The clock rates in S and S’ are considered to be the same. How does the relative velocity affect the way that an observer in S measures the time interval between flashes in S’?

 

The distance of S’ from S increases by the amount vt_d between flashes if S’ is traveling directly away from S. Light will have that additional distance to travel, and each successive flash will take vt_d/c longer to reach S. The observer on S would therefore measure the time interval as t_d’ = t_d + vt_d/c = t_d(1 + v/c), and it would appear to the observer that the clock rate on S’ is slower than that on S. If S’ were coming directly toward S, then the interval difference would be t_d’ = t_d(1 – v/c) and the clock rate would appear to be faster.

 

If S’ is not moving directly away from S, then the clock rate would appear to get progressively slower at a decreasing rate as the distance of S’ from S increases. Figure 1 illustrates this.

 

 

 

S is in the center of the circles. The straight line represents the path of S’. The circles intersect the path at the point where the device flashes. The dotted lines represent the path that light takes to reach the observer on S from S’. The additional time it takes the light to reach S after each flash is the difference in the radius of the circle for that flash and the radius of the circle for the previous flash. As can be seen, the radius increase becomes greater as the distance of S’ from S increases, and approaches the value of t_d’. It is assumed that the clock rate in the space between S and S’ remains constant. It is clear that relative velocity can make it appear that there is a change in clock rate when making intra-system measurements.

 

Experimental evidence would seem to indicate that time really does change as a function of velocity. Is there any reason why this relationship should exist? Although it is accepted that time can indeed vary, it seems highly unlikely that it would vary for more than one reason. We know that it varies as a function of distance from a physical body such as the earth. Is there any relationship between that and the velocity of a physical body? The following line of reasoning suggests that there is.

 

The variation of time would appear to be associated with the containment or confinement of energy. Energy is certainly contained when it is in the form of matter. Some cosmological observations seem to indicate that certain regions that contain large amounts of energy but little matter show evidence of more gravitational attraction than could be attributed to the matter present. As discussed elsewhere, gravitational attraction is equivalent to a time gradient – the greater the attraction, the denser the time gradient or the slower the time. This relationship of time to the confinement of energy suggests why time could vary as a function of velocity. As the velocity of an object increases, its kinetic energy increases. This kinetic energy is confined to the object, at least until the object collides with something. With more confined energy associated with the object, the time gradient associated with the object is steeper, which implies a slower clock rate. It is of interest to look at the ratio of the rest energy associated with an object and the overall energy associated with it when it has some velocity v, which results in the following formula:

 

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An object will have some time gradient associated with it as a result of the properties of the object itself, and thus some clock rate will exist at the surface of the object. Does the density of the time gradient through which an object is moving have any effect on the change in clock rate, if any, that is a function of the velocity of the object[3]? If the answer to these questions is yes, then there is probably not any single conversion factor that is appropriate to all situations, or that some additional factor is necessary such as some function of G.

 

It seems reasonable that time does vary as a function of velocity, but not necessarily as a function of velocity relative to some other object. It is not clear exactly what the functional relationship of time to velocity is. Although the factor γ from the Lorentz transformation is frequently used to convert time, that is not the actual conversion that is indicated by the transformation equation (there is a numerator factor present also). Most reasonable factors converge at very low values of v. Those factors that are similar to  or γ (as opposed to , etc.) also converge for very high values of v. Distinguishing between such factors on the basis of experimental data is difficult, since it appears that most experimental data is obtained with velocities near these extreme values.

 

In view of this, the following time factors will be used when necessary in the examples. It appears that the same general conclusions would be reached when using other reasonable time factors. Since distance is considered in these discussions to be invariant in different systems, we can write the following relationship, where c and t are the speed of light and a time interval in system S, and c’ and t’ are the equivalent in system S’:

 

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If we assume that S’ has some velocity v, and that

 

 

is the appropriate conversion (or correction) factor for time, then

 

 

which gives

 

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The conversion of the velocity v from the clock rate in S to that in S’ is

 

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A simpler view suggested by the theorem of the addition of velocities is that . This gives the time conversion as

 

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and the velocity conversion as

 

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These are two possible factors that will be used when necessary in a description or analysis of some event or events. Using either factor will generally provide equivalent results at very low or very high values of v. The former will be called the gamma factor, and the latter the basic factor. Note: results subsequent to writing this suggest that the gamma factor is probably the correct one for time conversions.